Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | Dic3×#126 |
| Sylow | C3 ⋊ C4×#126 |
516 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,15)(16,18,19,23)(17,21,20,22);(8,10,11,14)(9,12,13,15)(17,20)(21,22);(8,9)(10,12)(11,13)(14,15)(16,17)(18,22)(19,20)(21,23);(2,3)(4,5,6,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×#126, C2×Dic3×#126, C4×#5706, … |
| maximal normal subgroup of | C4×D3×#126, C2×Dic3×#126, C4×#5706, … |
| maximal quotient of | Dic3×#388, Dic3×#387, Dic3×#383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15649);